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Appears in collection : Jean-Morlet Chair 2022 - Conference: Nonlinear PDEs in Fluid Dynamics / Chaire Jean-Morlet 2022 - Conférence : EDP non-linéaires en dynamique des fluides

I will consider the motion of an incompressible viscous fluid on compact surfaces without boundary. Local in time well-posedness is established in the framework of $L_{p}$-$L_{q}$ maximal regularity for initial values in critical spaces. It will be shown that the set of equilibria consists exactly of the Killing vector fields. Each equilibrium is stable and any solution starting close to an equilibrium converges at an exponential rate to a (possibly different) equilibrium. In case the surface is two-dimensional, it will be shown that any solution with divergence free initial value in $L_{2}$ exists globally and converges to an equilibrium.

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Citation data

  • DOI 10.24350/CIRM.V.19917103
  • Cite this video Simonett, Gieri (09/05/2022). On the Navier-Stokes equations on surfaces. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19917103
  • URL https://dx.doi.org/10.24350/CIRM.V.19917103

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Bibliography

  • SIMONETT, Gieri et WILKE, Mathias. $ H^\infty $-calculus for the surface Stokes operator and applications. arXiv preprint arXiv:2111.12586, 2021. - https://arxiv.org/abs/2111.12586
  • PRÜSS, Jan, SIMONETT, Gieri, et WILKE, Mathias. On the Navier–Stokes equations on surfaces. Journal of Evolution Equations, 2021, vol. 21, no 3, p. 3153-3179. - http://dx.doi.org/10.1007/s00028-020-00648-0
  • PRÜSS, Jan, SIMONETT, Gieri, et WILKE, Mathias. Critical spaces for quasilinear parabolic evolution equations and applications. Journal of Differential Equations, 2018, vol. 264, no 3, p. 2028-2074. - https://doi.org/10.1016/j.jde.2017.10.010
  • PRÜSS, Jan et SIMONETT, Gieri. Moving interfaces and quasilinear parabolic evolution equations. Basel : Birkhäuser, 2016. - http://dx.doi.org/10.1007/978-3-319-27698-4

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